A performance characteristic of M-ary phase shift keying (MPSK) receivers is the variance of the phase error between the received and recovered signal carriers. For binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) loops utilizing integrate and dump filters and operating in the linear region, closed-form solutions for this variance exist (Hinedi and Lindsey 1989; and Yuen 1983). In this paper the variance is found by numerical methods for M>4. For verification and to investigate operation in the nonlinear region, computer simulation and hardware modeling were used (Kopp 1994).

The exponential bound parameter is numerically evaluated for 16-zone quantized 8-PSK for signal vectors bisecting decision regions and for signal vectors lying on decision boundaries. For decision regions of equal span, signal vectors lying on decision boundaries are superior, confirming the result of Parsons and Wilson (1990). Signal vectors bisecting decision regions are superior if span of decision regions not containing signal vectors is optimized. Using the partial derivative, it is shown that the optimal configuration depends on SNR. Furthermore, the optimal configuration is not necessarily one in which the decision boundaries are straight lines.>

TCM, combining modulation and coding, achieves coding gains over conventional uncoded multilevel modulation without the attendant bandwidth expansion. Since TCM was proposed Ungerboeck (1982, 1987) substantial work has done in this area. A large portion of the TCM work has been in the area of high-speed data transmission over voice grade modems using quadrature amplitude modulation, QAM. QAM, not having a constant envelope, is unattractive for employing a TWT with its nonlinear behavior as the power stage. Additional work has been done in utilizing M-ary PSK with TCM. Simulations by Taylor and Chan (1981) utilizing a 4-state convolutional code demonstrated the coding gain of a rate 2/3 coded 8-PSK modulation scheme. Wilson et. al. (1984) obtained results for 16-PSK TCM using codes with 4 to 32 states and achieved coding gains of 3.5 to 4.8 dB respectively, over 8-PSK and demonstrated that small memory codes achieved good gains with simple design procedures.>

Continuous phase frequency shift keying (CPFSK) is potentially an attractive modulation scheme for use on channels whose performance is limited by thermal noise. In this paper results for the performance available with CPFSK are given for coherent detection and noncoherent detection with arbitrary modulation indices and arbitrary observation intervals. This work serves two purposes. First, it provides interesting, new results for the noncoherent detection of CPFSK which indicate that the performance of such a system can be better than the performance of coherent PSK. Secondly, it provides a complete analysis of the performance of CPFSK at high SNR as well as low SNR and thereby unifies and extends the results previously available.